Sampling Distributions

Chapter 8

Sampling Distributions

1


Introduction
Generally, we are interested in population
parameters.
When the census is impossible, we draw a sample
from the population, then construct sample statistics,
that have close relationship to the population
parameters.

2


Introduction
Samples are random, so the sample statistic is a
random variable.
As such it has a sampling distribution.

3


8.1 Sampling Distribution of the Mean
Example 1: A die is thrown infinitely many times. Let
X represent the number of spots showing on any
throw. The probability distribution of X is
x

1 2 3 4 5 6
p(x) 1/6 1/6 1/6 1/6 1/6 1/6

E(X) = 1(1/6) +2(1/6) + 3(1/6)+………………….=
3.5
4
V(X) = (1-3.5)2(1/6) + (2-3.5)2(1/6) +….……. …= 2.92


Suppose we want to estimate  from the mean
of a sample of size n = 2.
What is the distribution of x ?

5


Throwing a die twice – sample mean
these
are the
means
of each
These are And
all the
possible
pairs
of values
for pair
the 2 throws

6

The distribution of

x when n = 2

Calculating the relative frequency of each value
of x we have the following results

1
Frequency1
1/36
Relative freq
(1+1)/2 = 1

1.5
2

2.0
3

2/36 3/36

2.5
4
4/36

(1+2)/2 = 1.5
(2+1)/2 = 1.5

3.0
5
5/36

3.5
6
6/36

(1+3)/2 = 2
(2+2)/2 = 2
(3+1)/2 = 2

4.0
5
5/36

4.5
4
4/36

5.0
3

5.5 6.0
2
1

3/36

2/36

1/36

Notice there are 36 possible
pairs of values:
1,1 1,2 ….. 1,6
2,1 2,2 ….. 2,6
………………..
6,1 6,2 ….. 6,6

7


n 5

n 10

n 25

 x 3.5

 x 3.5

 x 3.5

 2x
 .5833 (  )
5

 2x

2
 x .2917 (  )
10

 2x
 .1167 (  )
25

2
x

2
x

8

As the sample size changes, the mean of the
sample mean does not change!


n 5

n 10

n 25

 x 3.5

 x 3.5

 x 3.5

 2x
 .5833 (  )
5

 2x
2
 x .2917 (  )
10

 2x
 .1167 (  )
25

2
x

2
x

9

As the sample size increases, the
variance of the sample mean decreases!


Demonstration: Why is the variance of the
sample mean is smaller than the population
variance?

Mean = 1.5 Mean = 2. Mean = 2.5

Population

1

1.5

2

2.5

3

Compare
thetake
range
of the population
Let us
samples
to the
range
of the sample mean.
of two
observations.

10

The Central Limit Theorem
If a random sample is drawn from any population, the
sampling distribution of the sample mean is:
– Normal if the parent population is normal,
– Approximately normal if the parent population is
not normal, provided the sample size is
sufficiently large. The larger the sample size, the
more closely the sampling distribution of x will
resemble a normal distribution.

11


The mean of X is equal to the mean of the parent
population

μ x μx

The variance of X is equal to the parent population
variance divided by ‘n’.
2
x

σ
σ 
n
2
x

12

n

Sampling Distribution

1

Population distribution

2

5

30


30

50

70

90

120

Census

n Sampling Distribution
1 Normal
Pop distribution

2

5

3
0


Example 2: The amount of soda pop in each bottle is
normally distributed with a mean of 32.2 ounces and
a standard deviation of .3 ounces.
Find the probability that a bottle bought by a
customer will contain more than 32 ounces.
0.7486

P(x  32)
x = 32  = 32.2

x  μ 32  32.2
P(x  32) P(

) P(z   .67) 0.7486
σx
.3
16

Find the probability that a carton of four bottles will
have a mean of more than 32 ounces of soda per
bottle.

x   32  32.2
P( x  32) P(

)
x
.3 4
P( z   1.33 ) 0.9082

P(x  32)

x 32  x 32.2

17


Example 3: The average weekly income of B.B.A
graduates one year after graduation is $600. Suppose
the distribution of weekly income has a standard
deviation of $100.
What is the probability that 35 randomly selected
graduates have an average weekly income of less than
$550?

x  μ 550  600
P(x  550) P(


)
σx
100 35
P(z   2.97) 0.0015
18


19